- #1

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Suppose your operator is M, and the matrix element between two position states is <x|M|y>=M(x,y).

It seems that, at least formally, any function f(x,y) can be written as an infinite sum of deltas and derivative of deltas, with constant coefficients. So for example, the Green's function which is [itex]\langle x|(\partial^2-m^2)^{-1} |y\rangle=\text{BesselFunction}(x-y)[/itex] can be written as an infinite sum of deltas and derivatives of deltas, but not a finite sum, so it's not local.

But then what about functions of the momentum operator, such as [itex]\log[P] [/itex] or [itex]e^{iP} [/itex]? Are these local?

Is those functions aren't local, is it safe to say that only polynomials in the momentum operator can be local?

Also, the position operator squared has this matrix element: <x|X

^{2}|y>=δ(x-y)x

^{2}. If we wanted to expand this matrix element as a sum of delta functions and derivatives with constant coefficients, it would require an infinite amount. However, if we let the coefficients depend on x, then it is a finite amount.

So should the definition of a local operator be modified to as any operator whose matrix elements in position space is a finite sum of delta functions and derivatives of delta functions with coefficients depending only on position?